三次样条插值c语言自然边界条件

在三次样条插值中，自然边界条件是指首尾两个数据点的二阶导数为零。以下是一个使用C语言实现带有自然边界条件的三次样条插值的示例代码：

#include <stdio.h>

// 定义常量
#define N 5
#define SIZE (N + 1)

// 定义数据点结构体
typedef struct {
    double x;
    double y;
} Point;

// 计算三次样条插值的系数
void calculateCoefficients(Point points[], double h[], double alpha[], double l[], double miu[], double z[]) {
    int i;

    // 初始化辅助数组
    alpha[0] = 0;
    l[0] = 1;
    miu[0] = 0;
    z[0] = 0;

    // 计算中间系数
    for (i = 1; i < N; i++) {
        double A = h[i-1];
        double B = 2 * (h[i-1] + h[i]);
        double C = h[i];
        double D = 6 * ((points[i+1].y - points[i].y) / h[i] - (points[i].y - points[i-1].y) / h[i-1]);

        alpha[i] = C / (B - alpha[i-1] * A);
        l[i] = (A * l[i-1] + D) / (B - alpha[i-1] * A);
        miu[i] = -C / (B - alpha[i-1] * A);
        z[i] = (D - alpha[i-1] * z[i-1]) / (B - alpha[i-1] * A);
    }

    // 初始化最后一个辅助数组
    l[N] = 1;
    z[N] = 0;

    // 设置自然边界条件
    z[0] = 0;
    z[N] = 0;
}

// 使用三次样条插值估计给定位置的数值
double interpolate(Point points[], double h[], double alpha[], double l[], double miu[], double z[], double target) {
    int i;

    // 在给定位置进行插值
    for (i = N - 1; i >= 0; i--) {
        if (target >= points[i].x && target <= points[i+1].x) {
            double t = (target - points[i].x) / h[i];
            double a = (1 - t) * points[i].y;
            double b = t * points[i+1].y;
            double c = ((t * t * t - t) * z[i] + (t - t * t * t) * z[i+1]) * (h[i] * h[i]) / 6;

            return a + b + c;
        }
    }

    return 0;
}

int main() {
    // 定义数据点
    Point points[SIZE] = {
        {0, 2},
        {1, 3},
        {2, 5},
        {3, 4},
        {4, 1},
        {5, 6}
    };

    // 计算步长
    double h[SIZE];
    int i;
    for (i = 0; i < N; i++) {
        h[i] = points[i+1].x - points[i].x;
    }

    // 定义辅助数组
    double alpha[SIZE], l[SIZE], miu[SIZE], z[SIZE];

    // 计算系数
    calculateCoefficients(po